Some new sequence spaces defined by lacunary sequences

DOI: 10.2478/s12175-007-0034-3 Math. Slovaca 57 (2007), No. 4, 393–399

SOME NEW SEQUENCE SPACES DEFINED BY LACUNARY SEQUENCES

Ekrem Savas* — Vatan Karakaya**

(Communicated by Pavel Kostyrko )

ABSTRACT. It is natural to expect that lacunary almost convergence must be related to the some concept of lacunary almost bounded variations in the some view as almost convergence is related to almost bounded variation. The purpose of this paper is to examine this new concept in some details. Some inclusion theorems have been established.

2007c Mathematical Institute Slovak Academy of Sciences

1. Introduction

Let s be the set of all sequences real and complex and
∞, c and c0respectively be the Banach spaces of bounded, convergent and null sequences x = (xn) normed by
x
= sup

n |xn|. Let D be the shift operator on s. That is, Dx = {xn}^∞_n=1, D²x = {xn}^∞_n=2, . . .

and so on. It is evident that D is a bounded linear operator on
∞ onto itself and that

D^k

= 1 for every k.

It may be recalled that Banach limit L is a non-negative linear functional on

∞ such that L is invariant under the shift operator, that is, L (Dx) = L (x) for all x ∈
∞ and that L (e) = 1 where e = (1, 1, 1, . . . ), (see, B a n a c h [1]).

A sequence x ∈
∞ is called almost convergent (see, L o r e n t z [2]) if all Banach Limits of x coincide.

Let ˆc denote the set of all almost convergent sequences. L o r e n t z [2] proved that

ˆc =

 x : lim

m dmn(x) exists uniformly in n



2000 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n: Primary 40A05.

K e y w o r d s: lacunary sequences, almost convergence, almost bounded variation.

where

dmn(x) = xn+ xn+1+· · · + xn+m

m + 1

Let θ = (kr) be the sequence of positive integers such that i) k0= 0 and 0 < kr < kr+1

ii) hr= (kr− kr−1)→ ∞ as r → ∞.

Then θ is called a lacuanry sequence. The intervals determined by θ are denoted by Ir = (kr−1, kr]. The ratio _k^kr

r−1 will be denoted by qr (see, F r e e d m a n et al [5]).

Recently, D a s and M i s h r a [3] defined Mθ, the set of almost lacunary convergent sequences, as follows:

Mθ=



x : there exists l such that uniformly in i ≥ 0,

r→∞lim

h1r



k∈Ir

(xk+i− l) = 0 .

It is natural to expect that lacunary almost convergence must be related to some concept BV^∧ θ in the same view as almost convergence is related to the concept ofBV .^∧ BV denotes the set of all sequences of almost bounded variation and a^∧ sequence inBV^∧ θ will mean a sequence of lacunary almost bounded variation.

The main object of this paper is to study this new concept in some details.

Also a new sequence space

b∧

BVθ which is apparently more generals than BV^∧ θ

naturally comes up for investigation and is considered along withBV^∧ θ.

We may remark here that the concept BV of almost bounded variation^∧ have been recently introduced and investigated by N a n d a and N a y a k [4] as follows:

∧

BV =

 x : 

m |tmn(x)| converges uniformly in n



where

tmn(x) = 1 m (m + 1)

m v=1

v (xn+v− xn+v−1) . Put

trn= trn(x) = 1 hr+ 1

kr+1 j=kr−1+1

xj+n. Then write r, n > 0,

ϕrn(x) = trn(x) − tr−1n(x) .

When r > 1, straightforward calculation shows that ϕrn(x) = ϕrn= 1

hr(hr+ 1)

hr



u=1

u

xkr−1+u+1+n− xkr−1+u+n . Now write

∧

BVθ =

 x : 

r |ϕrn(x)| converges uniformly in n



and

∧b

BVθ=



x : sup

n



r |ϕrn(x)| < ∞

 .

Here and afterwards summation without limits sum from 1 to∞.

2. Main results

We have the following theorem.

 1^ BV^∧θ⊂BV^∧b θ for every θ.

P r o o f. Suppose that x ∈ BV^∧θ and write ϕrn for ϕrn(x). We have to show that 

r |ϕrn| is bounded. By the definition of BV^∧θ there exists an integer R such that, for all n,



r≥R+1

|ϕrn| ≤ 1.

Therefore it follows that for r ≥ R + 1, and all n

|ϕrn| ≤ 1.

It is enough to show that, for fixed r, ϕrn is bounded in n. Let r ≥ 2 be fixed. A straightforward calculation shows that

xkr+1+n− xkr+n = (hr+ 1) ϕrn− (hr− 1) ϕr−1n.

Hence for any fixed r > R+1, xkr+1+n−xkr+nis bounded and so ϕrnis bounded for all r and n.

This completes the proof.

Remark 1^ It is now a pertinent question, whether

b∧

BVθ⊂BV^∧θ, that is whether BV∧θ=

∧b

BVθ. We are not able to answer this question and it remains open.

We have:

 2^ BV^∧θ is Banach space normed by

x
= sup

n



r

|ϕrn| (2.1)

P r o o f. Because of Theorem 1, (2.1) is meaningful. It is routine verification that BV^∧θ is a normed linear space. To show that BV^∧θ is complete in its norm topology, let

xⁱ_∞

i=0 be a Cauchy sequence in BV^∧θ. Then xⁱ_n∞

i=0 is a Cauchy sequence inC for each n. Therefore xⁱ_n → xn (say). Put x = {xn}^∞_i=0. We now show that x ∈BV^∧θ and

xⁱ− x

→0. Since

xⁱ

is a Cauchy sequence inBV^∧θ, given ε > 0, there exists N such that for i, j ≥ N ,



r

ϕrn

xⁱ− x^j < ε

for all n. Therefore for any M and i, j ≥ N ;

M r=0

ϕrn

xⁱ− x^j < ε

for all n. Now taking limit as j → ∞ and then as M → ∞, we get for i > N



r

ϕrn

xⁱ− x < ε (2.2)

for all n.

Thus xⁱ− x ∈ BV^∧θ and therefore by linearity x ∈ BV^∧θ. Also (2.2) implies that

xⁱ− x

< ε (i ≥ N ).

This completes the proof.

 3^ If lim inf qr > 1, BV ⊂^∧ BV^∧θ.

P r o o f. Let x ∈BV . Since lim inf q^∧ r> 1, qr > 1 + δ for δ > 0. Now we have 1

hr(hr+ 1)

kr



j=kr−1+1

(j − kr−1) (xj+n+1− xj+n)

= 1

hr(hr+ 1)

kr



j=kr−1+1

j (xj+n+1− xj+n)

− kr−1

hr(hr+ 1)

kr



j=kr−1+1

j (xj+n+1− xj+n)

≤ 1

hr(hr+ 1)

kr



j=kr−1+1

j (xj+n+1− xj+n) .

By using property of lacunary sequence, we have 1

hr(hr+ 1)

kr



j=kr−1+1

j (xj+n+1− xj+n)

= 1

hr(hr+ 1)

⎡

⎣^kr

j=1

j (xj+n+1− xj+n)−

kr−1

j=1

j (xj+n+1− xj+n)

⎤

⎦

= (kr+ 1) kr

hr(hr+ 1) 1 (kr+ 1) kr

kr



j=1

j (xj+n+1− xj+n)

−(kr−1+ 1) kr−1

hr(hr+ 1)

1 (kr−1+ 1) kr−1

kr−1

j=1

j (xj+n+1− xj+n)

We now have

(kr+ 1) kr

hr(hr+ 1) = kr

hr

kr+ 1 hr+ 1



= kr

hr

 kr+ 1 kr− kr−1+ 1



= kr

hr

 1

kr−kr−1+1 kr+1



= kr

hr

 1 1− _k^k_r^r−1₊₁

 .

Since kr< kr+ 1 and 1− _k^k_r^r−1₊₁ > 0, we have (kr+ 1) kr

hr(hr+ 1) ≤ kr

hr

 1 1− ^kr−1_kr



=

 1 1−^kr−1_kr

  1 1− ^kr−1_kr

 .

Since qr> 1 + δ, we get

(kr+ 1) kr

hr(hr+ 1) ≤

1 + δ δ

2

. Again

(kr−1+ 1) kr−1

hr(hr+ 1) = kr−1

hr

kr−1+ 1 hr+ 1



= kr−1

hr

 kr−1+ 1 kr− kr−1+ 1



= kr−1

hr

 1

kr−kr−1+1 kr−1+1



= kr−1

hr

 1

kr+2 kr−1+1− 1

 .

Since kr−1+ 1 < kr+ 2 for every r ∈ N, 1 <_k^k_r−1^r+2₊₁. Hence we obtain (kr−1+ 1) kr−1

hr(hr+ 1) = kr−1

hr

 1

kr+2 kr−1+1− 1



=

 1

kr+2 kr−1+1 − 1

  1

kr

kr−1 − 1



≤

1 δ

2

.

Consequently, it can be seen that, for all n and r ≥ 2,



r

  1

hr(hr+ 1)

kr



j=kr−1+1

(j − kr−1) (xj+n+1− xj+n)  

≤

1 + δ δ

2

r

  1

kr(kr+ 1)

kr



j=1

j (xj+n+1− xj+n)   +

1 δ

2

r

 1

kr−1(kr−1+ 1)

kr−1

j=1

j (xj+n+1− xj+n)  .

Since each of sums on left converges to any limit uniformly in n, the sum on right converges also to any limit uniformly in n. So we get x ∈BV^∧θ. This completes

the proof.

REFERENCES

[1] BANACH, S.: Theorie des operations linearies, Warszawa, 1932.

[2] LORENTZ, G. G.: A contribution to the theory of divergent series, Acta Math.80 (1948), 167–190.

[3] DAS, D.—MISHRA, S. K.: Banach limits and lacunary strong almost convergence, J. Orissa Math. Soc.2 (1983), 61–70.

[4] NANDA, S.—NAYAK, K. C.: Some new sequence spaces, Indian J. Pure Appl. Math.9 (1978), 836–846.

[5] FREEDMAN, A. R.—SEMBER, J. J.—RAPHEAL, M.: Some Cesaro-type summability spacces, Proc. London Math. Soc. (3)37 (1973), 508–520.

Received 14. 5. 2004 * Department of Mathematics

˙Istanbul Ticaret University Uskudar, ˙Istanbul

TURKEY

E-mail : ekremsavas@yahoo.com

** Department of Mathematics Education Faculty of Adıyaman Gaziantep University

Adıyaman TURKEY

E-mail : vkkaya@yahoo.com